CF - Santa's Bot

Say we had two fractions $x=\frac{a}{b}$ and $y=\frac{c}{d}$. Let $f$ be a function that calculates the "total" of a fraction (that is, the numerator times the modular inverse of the denominator). To find $f(x + y)$, we can simply calculate $f(x) + f(y)$. Thus, to find the total of the overall probability, we can just add up all the totals of all the smaller probabilities, keeping a running sum as we go.